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<h1 class="reftitle">interiorPoint</h1>
<h2>Purpose</h2>
<p>Compute a point in the relative interior of the Polyhedron.</p>
<h2>Syntax</h2>
<pre class="synopsis">sol = P.interiorPoint</pre>
<pre class="synopsis">sol = P.interiorPoint(facetIndex)</pre>
<h2>Description</h2>
<p></p>
	  Compute a point in the relative interior of the polyhedron. 

	  If <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint1.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint1.png"> is specified, then a point
	  in the relative interior of <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint2.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint2.png"> is returned.
	<h2>Input Arguments</h2>
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<td><tt>P</tt></td>
<td>
<p></p>Polyhedron in any format<p>
	    		Class: <tt>Polyhedron</tt></p>
</td>
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<td><tt>facetIndex</tt></td>
<td>
<p></p>Index of an inequality of <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint3.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint3.png"> (row of <tt>P.H</tt>).<p>
	    		Class: <tt>integer</tt></p>
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<h2>Output Arguments</h2>
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<td><tt>sol</tt></td>
<td>
<p></p>
<p>
	    		Class: <tt>struct</tt><p></p><tr valign="top">
<td><tt>sol.x</tt></td>
<td>
<p></p>The interior point<p>
	    		Class: <tt>double vector</tt></p>
</td>
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<td><tt>sol.isStrict</tt></td>
<td>
<p></p>The output is true if <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint4.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint4.png"> is in the strict relative interior, false
                otherwise.<p>
	    		Class: <tt>logical</tt></p>
</td>
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<td><tt>sol.r</tt></td>
<td>
<p></p>Radius of the largest ball centered at <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint5.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint5.png"> that is still within <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint6.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint6.png">
                    
                    
               <p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint9.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint9.png"></p>
                    
                Note : <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint7.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint7.png"> is empty if <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint8.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint8.png"> is empty or only has a V-rep.<p>
	    		Class: <tt>double</tt></p>
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<h2>Example(s)</h2>
<h3>Example 
				1</h3>Compute interior point in unbounded polyhedra<pre class="programlisting">P = Polyhedron('V', randn(20,3), 'R', -[1 0 0]);</pre>
<pre class="programlisting"></pre>
<pre class="programlisting">sol = P.interiorPoint</pre>
<pre class="programlisting">
sol = 

           x: [3x1 double]
    isStrict: 1
           r: []

</pre>
<h3>Example 
				2</h3>Compute a point in the relative interior of the fourth facet<pre class="programlisting">P = Polyhedron('H',[sin([0:0.5:2*pi])' cos([0:0.5:2*pi])' ones(13,1)]);</pre>
<pre class="programlisting"></pre> Polyhedron must be in its minimal representation to compute facets. Perform redundancy elimination.<pre class="programlisting"> P.minHRep(); </pre>
<pre class="programlisting"></pre> Compute the center of the fourth facet <pre class="programlisting">sol = P.interiorPoint(4)</pre>
<pre class="programlisting">
sol = 

           x: [2x1 double]
    isStrict: 1
           r: 0.122417438109627

</pre>
<pre class="programlisting">plot(P); hold on;
		pplot(sol.x, 'ro', 'markerfacecolor', 'r', 'markersize', 10); </pre>
<pre class="programlisting"></pre>
<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint_img_1.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint_img_1.png" width="60%"></p>
<h3>Example 
				3</h3>Compute a point in the relative interior of a lower-dimensional polyhedron.<pre class="programlisting">P = Polyhedron('H',[randn(20,3) ones(20,1)], 'He', [0 0 1 0]);</pre>
<pre class="programlisting"></pre>
<pre class="programlisting">sol = P.interiorPoint</pre>
<pre class="programlisting">
sol = 

           x: [3x1 double]
    isStrict: 0
           r: 0.39490124253373

</pre>
<pre class="programlisting">plot(P); hold on;
		pplot(sol.x, 'ro', 'markerfacecolor', 'r', 'markersize', 10); </pre>
<pre class="programlisting"></pre>
<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint_img_2.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint_img_2.png" width="60%"></p>
<h2>See Also</h2>
<a href="./chebycenter.html">chebycenter</a>, <a href="./facetinteriorpoints.html">facetinteriorpoints</a><p></p>
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<br><p>©  <b>2010-2013</b>     Colin Neil Jones: EPF Lausanne,    <a href="mailto:colin.jones@epfl.ch">colin.jones@epfl.ch</a></p>
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